A local projection stabilization finite element method with nonlinear crosswind diffusion for convection-diffusion-reaction equations

DOI:10.1051/m2an/2013071 www.esaim-m2an.org

A LOCAL PROJECTION STABILIZATION FINITE ELEMENT METHOD WITH NONLINEAR CROSSWIND DIFFUSION FOR

CONVECTION-DIFFUSION-REACTION EQUATIONS

Gabriel R. Barrenechea

, Volker John

and Petr Knobloch

Abstract. An extension of the local projection stabilization (LPS) finite element method for convection-diffusion-reaction equations is presented and analyzed, both in the steady-state and the transient setting. In addition to the standard LPS method, a nonlinear crosswind diffusion term is introduced that accounts for the reduction of spurious oscillations. The existence of a solution can be proved and, depending on the choice of the stabilization parameter, also its uniqueness. Error estimates are derived which are supported by numerical studies. These studies demonstrate also the reduction of the spurious oscillations.

Mathematics Subject Classification. 65N30, 65N12, 65N15, 65M60.

Received October 30, 2012. Revised February 8, 2013.

Published online July 30, 2013.

1. Introduction

The solution of convection-dominated convection-diffusion-reaction equations with finite element methods constitutes a very challenging (and open) problem. Over the last three decades, the amount of work devoted to this problem is impressive. The usual way of treating dominating convection, at least in the context of finite element methods, consists in adding extra terms to the standard Galerkin formulation, aimed at enhancing the stability of the discrete solution by means of introducing artificial diffusion. These new terms vary according to the method, and can be residual-based, as in the SUPG/GLS/SDFEM family (see [6,13,14,16,29]), or edge based, such as the CIP method (see [7,9]). For an up-to-date and thorough review of these and other techniques, see [31]. It is striking to notice that, despite the impressive amount of work that has been devoted to this topic, up to now there is not a method that ‘ticks all the boxes’, i.e., a method that produces sharp layers while avoiding oscillations, see [1] for a recent review and a numerical assessment.

Keywords and phrases.Finite element method, local projection stabilization, crosswind diffusion, convection-diffusion-reaction equation, well posedness, time dependent problem, stability, error estimates.

1 Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland.

[email protected]

2 Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany and Free University of Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany.[email protected]

3 Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University, Sokolovsk´a 83, 18675 Praha 8, Czech Republic.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

Among the various stabilized finite element methods, the local projection stabilization (LPS) method has received some attention over the last decade. Originally proposed for the Stokes problem in [2], and extended to the Oseen equations in [4] (see also [5,30]), the LPS method has been also used recently to treat convection- diffusion equations (see [15,24–26]). The basic idea of this method consists in restricting the direct application of the stabilization to so-called fluctuations or resolved small scales, which are defined by local projections. It has several attractive features, such as adding symmetric terms to the formulation and avoiding the computation of second derivatives of the basis functions (thus using only information that is needed for the assembly of the matrices from the standard Galerkin method). Unfortunately, the solutions obtained with the LPS method possess the same deficiency like solutions computed, e.g., with the SUPG method: non-negligible spurious oscillations are often present in a vicinity of layers.

Motivated by the wish of recovering the monotonicity properties of the continuous problem, which might be crucial in applications, a number of so-called Spurious Oscillations at Layers Diminishing (SOLD) methods were proposed. SOLD methods add an extra term to the already stabilized formulation, which usually depends on the discrete solution in a nonlinear way, vanishes for small residuals (thus acting mostly at layers), and adds some extra, but different, diffusivity to the formulation. In particular, methods that add crosswind diffu- sion, like the one proposed in [11], have been proved to belong to the best SOLD methods in comprehensive studies [17,18]. Although these methods diminish oscillations considerably, no single method succeeds to fully eliminate them [17,18,23]. Also, from a purely mathematical point of view, it is unknown if these methods lead to well-posed problems. In fact, existence of solutions is usually possible to prove, but, to our best knowledge, there is no nonlinear SOLD method that is known to produce a unique solution, see [7,27] for a discussion of this topic.

Based on the previous considerations, this paper has three major objectives, namely:

• to improve the quality of the LPS solution (especially in the vicinity of layers);

• to explore the applicability of SOLD-type strategies within a LPS context; and

• to contribute to the mathematical understanding of nonlinear stabilization techniques for the convection- diffusion equation.

Hence, in this work we propose a LPS method with nonlinear crosswind diffusion for convection-diffusion- reaction equations. Two ways for choosing the parameter in the crosswind diffusion term will be studied. The first choice uses global information obtained from the data of the problem, whereas the second proposal is completely local, employing information of the computed solution instead of the data. For the first approach, which is the simpler one, the existence and the uniqueness of the solution can be proved for the steady-state and time-dependent equations, where the latter is discretized in time with an implicit one-stepθ-scheme. To our best knowledge, this is the first nonlinear discretization for convection-diffusion-reaction equations for which both, existence and uniqueness of a solution can be shown. The form of the crosswind term resembles the Smagorinsky Large Eddy Simulation (LES) model which was analyzed in [28]. It involves fluctuations of a term mimicking ap-Laplacian. The crucial analytical property for proving the uniqueness of the solution is the strong monotonicity of the corresponding operator. For the more complicated local definition of the parameter, the analysis will show the existence of a solution and its uniqueness for the time-dependent discretization in the case of sufficiently small time steps.

The analysis is performed for the model problems of linear steady-state and time-dependent convection- diffusion-reaction equations. Applying a nonlinear discretization scheme to a linear problem leads certainly to a considerable complication of the solution process and to an additional numerical cost. This latter aspect can be overcome in the transient regime by using a semi-implicit (linearized) approach that computes the stabilization parameter with the solution from the previous discrete time. With respect to the former aspect, it has to be mentioned that the most important motivation for studying discretizations that reduce spurious oscillations comes from the need to address applications that lead to nonlinear coupled systems of convection- diffusion-reaction equations as in [21]. It was demonstrated in [21] that the locally large spurious oscillations of the SUPG method might lead to a fast blow-up of the simulations, and hence the reduction of the spurious

oscillations is essential to perform simulations at all. Thus, the reduction of the oscillations at layers becomes a priority, even over computational cost. It should be noted that in many applications, like in [21], only interior or characteristic layers are present, such that a method for reducing the oscillations has to work properly in particular for these types of layers. Finally, it is worth mentioning that our final aim is to address applications that lead to such coupled problems. Since these problems are nonlinear, the use of a nonlinear stabilization usually does not result in a notable complication of the solution procedure.

The plan of the paper is as follows. In the remaining part of this introduction, the problems of interest are stated and some basic notations are given. Section 2will summarize the main abstract hypothesis imposed on the different partitions of the domain and the finite element spaces considered. Section 3presents the method for the steady-state case, for which well-posedness is analyzed in Section 3.1 and error estimates are proved in Section 3.2. In Section 4, the method for the time-dependent problem is presented. Well-posedness and stability are proved in Section4.1and error estimates in Section4.2. Since the analysis is based on the abstract framework from Section 2, Section 5 presents some concrete examples that fit into this framework. Finally, numerical illustrations that support the analytical results and which demonstrate the reduction of spurious oscillations are presented in Section6.

Throughout the paper, standard notations are used for Sobolev spaces and corresponding norms, see,e.g., [10].

In particular, given a measurable setD⊂R^d, the inner product in L²(D) or L²(D)^d is denoted by (·,·)D and the notation (·,·) is used instead of (·,·)Ω. The norm (seminorm) in W^m,p(D) will be denoted by · _m,p,D (| · |_m,p,D), with the convention · _m,D= · _m,2,D, and the same notation is used for scalar and vector-valued functions.

1.1. The problems of interest

LetΩ⊂R^d,d∈ {2,3}, be a bounded polygonal (polyhedral) domain with a Lipschitz-continuous boundary

∂Ω and let us consider the steady-state convection-diffusion-reaction equation

−ε Δu+b· ∇u+c u=f in Ω, u=u_b on∂Ω. (1) It is assumed thatεis a positive constant andb∈W^1,∞(Ω)^d,c∈L^∞(Ω), f ∈L²(Ω), andu_b∈H^1/2(∂Ω) are given functions satisfying

σ:=c−1

2∇ ·b≥σ₀>0 inΩ, (2)

whereσ₀ is a constant. Then the boundary value problem (1) has a unique solution inH¹(Ω).

The condition σ₀ > 0 is often used in the analysis of stabilized finite element methods for the numerical solution of (1), see,e.g., [31], but it limits the applications of the theory since many problems of interest involve solenoidal convective velocities and no zero-order terms, which leads toσ₀= 0. Unfortunately, it is not known how to prove optimal convergence results even for the underlying linear local projection stabilization without assuming σ₀ >0, although numerical results do not indicate any deterioration of the convergence rates when σ₀= 0. The analysis of the nonlinear term introduced in this paper does not require this assumption.

Besides the steady-state case, also the time-dependent convection-diffusion-reaction equation u_t−ε Δu+b· ∇u+c u=f in (0, T]×Ω,

u=u_b in [0, T]×∂Ω, u(0,·) =u₀inΩ,

⎫⎬

⎭ (3) will be considered. In (3), [0, T] is a finite time interval, ε is assumed to be a positive constant, b ∈ L^∞(0, T;W^1,∞(Ω)^d), c ∈ L^∞(0, T;L^∞(Ω)), f ∈ L²(0, T;L²(Ω)), u_b ∈ L²(0, T;H^1/2(∂Ω)), and u₀ ∈ H¹(Ω) denotes the initial condition. The functionσis defined analogously to (2) and the inequality (2) is assumed to hold for allt∈[0, T]. In this case, the conditionσ₀ >0 can be circumvented by considering instead of (3) an equivalent problem forv=ue^{−α t} which satisfiesσ₀>0 for sufficiently largeα.

2. Assumptions on approximation spaces and the set M

From now on, C, ˜C or ¯C denote generic constants which may take different values at different occurrences but are always independent of the dataε,b,c,f, andu_b, the constantσ₀, and the discretization parameters (h andδtin the following).

Given h > 0, let W_h ⊂ W^1,∞(Ω) be a finite-dimensional space approximating the space H¹(Ω) and set V_h = W_h ∩H₀¹(Ω). Next, let Mh be a set consisting of a finite number of open subsets M of Ω such that Ω=∪M∈MhM. It will be supposed that, for anyM ∈Mh,

card{M∈Mh; M ∩M =∅} ≤C, (4)

h_M := diam(M)≤C h, (5)

h_M ≤C h_M ∀M ∈Mh, M∩M =∅, (6)

h^d_M ≤Cmeas_d(M). (7)

The spaceW_h is assumed to satisfy the local inverse inequality

|vh|_1,M ≤C h⁻¹_M vh_0,M ∀v_h∈W_h, M∈Mh. (8) For any M ∈Mh, a finite-dimensional spaceD_M ⊂L^∞(M) is introduced. It is assumed that there exists a positive constantβ_LP independent ofhsuch that

v∈VsupM

(v, q)_M

v_0,M ≥β_LPq_0,M ∀q∈D_M, M ∈Mh, (9) whereV_M ={vh∈V_h; v_h= 0 inΩ\M}. This hypothesis will be needed in what follows for the construction of a special interpolation operator (see Lemma3.7below). Concrete examples of spacesW_h andD_M satisfying the assumptions formulated here will be presented in Section5.

Furthermore, for anyM ∈Mh, a finite-dimensional spaceG_M ⊂L^∞(M) withG_M ⊃D_M is introduced such

that ∂v_h

∂x_i

M

∈G_M ∀v_h∈W_h, i= 1, . . . , d, and it is assumed that, for anyp∈[1,∞], there is a constantC such that

q_0,p,M≤C h

dp−^d₂

M q_0,M ∀ q∈G_M, M ∈Mh. (10)

To characterize the approximation properties of the spaces W_h and D_M, it is assumed that there exist interpolation operatorsi_h∈L(C(Ω), W_h)∩L(C(Ω)∩H₀¹(Ω), V_h) andj_M ∈L(H¹(M), D_M),M ∈Mh, such that, for some constantsl∈NandC >0 and for any setM ∈Mh, it holds

|v−i_hv|_1,M+h⁻¹_M v−i_hv_0,M ≤C h^k_M|v|_k+1,M ∀v∈H^k+1(M), k= 1, . . . , l, (11) q−j_Mq_0,M ≤C h^k_M|q|_k,M ∀q∈H^k(M), k= 1, . . . , l. (12) In addition, it is assumed that, for anyp∈[1,6],

|v−i_hv|_1,p,M≤C h^k+

dp−^d₂

M |v|_k+1,M ∀v∈H^k+1(M), k= 1, . . . , l. (13)

3. A local projection discretization of the steady-state problem

The weak form of problem (1) is: findu∈H¹(Ω) such thatu=u_b on∂Ω and

a(u, v) = (f, v) ∀v∈H₀¹(Ω), (14)

where the bilinear formais given by

a(u, v) :=ε(∇u,∇v) + (b· ∇u, v) + (c u, v).

As it was mentioned in the introduction, the most often used approach to cure the instabilities of the Galerkin method consists in adding extra terms to the formulation. To build these additional terms for the method studied here, for anyM ∈Mh, a continuous linear projection operatorπ_M is introduced which maps the spaceL²(M) onto the spaceD_M. It is assumed that

πM_L(L²_(M),L²_(M))≤C ∀M ∈Mh. (15) E.g., ifπ_M is the orthogonalL² projection, thenC = 1. Using this operator, the fluctuation operatorκ_M :=

id−π_M is defined, whereidis the identity operator onL²(M). Then, clearly

κM_L(L²_(M),L²_(M))≤C ∀M ∈Mh. (16) Sinceκ_M vanishes onD_M, it follows from (16) and (12) that

κMq_0,M ≤C h^k_M|q|_k,M ∀ q∈H^k(M), M ∈Mh, k= 0, . . . , l. (17) An application ofκ_M to a vector-valued function means thatκ_M is applied component-wise.

For anyM ∈Mh, a constantbM ∈R^d is chosen such that

|bM| ≤ b_0,∞,M, b−bM_0,∞,M ≤C h_M|b|_1,∞,M, (18) where| · |denotes the Euclidean norm inR^d. A typical choice forbM is the value ofbat one point ofM, or the integral mean value ofboverM. In addition, a functionu_bh ∈W_his introduced such that its trace approximates the boundary conditionu_b.

We are now ready to present the finite element method to be studied: findu_h∈W_hsuch thatu_h−u_bh∈V_h and

a(u_h, v_h) +s_h(u_h, v_h) +d_h(u_h;u_h, v_h) = (f, v_h) ∀v_h∈V_h, (19) where

s_h(u, v) =

M∈Mh

τ_M (κ_M(bM· ∇u), κM(bM· ∇v))_M, d_h(w;u, v) =

M∈Mh

τ_M^sold(w)κ_M(P_M∇u), κM(P_M∇v) _M,

andP_M :R^d→R^d is the projection onto the line (plane) orthogonal (crosswind) to the vectorbM defined by

P_M =

⎧⎨

⎩

I−bM ⊗bM

|bM|² ifbM =0,

0 ifbM =0,

I being the identity tensor. The stabilization parameters are given by τ_M =τ₀ min

h_M

b_0,∞,M,h²_M ε

, (20)

τ_M^sold(u_h) = ˜τ_M(u_h)|κM(P_M∇uh)|,

where τ₀ is a positive constant and ˜τ_M is a non-negative function ofu_h and the data of (1). Note that the crosswind stabilization term is ofp-Laplacian type withp= 3.

It remains to specify the function ˜τ_M. First, inspired by the definition of s_h, where each term in the sum is bounded byτ₀h_M|bM| κM∇u_0,MκM∇v_0,M, we set ˜τ_M(u_h) =γ_M(u_h)h_M|bM| with a functionγ_M still depending on u_h and/or the data of (1). Second, the function γ_M has to be chosen in such a way that the discrete problem preserves the following scaling properties of the problem (1):

• if the dataε,b,c, andf are replaced byα ε,αb,α c, andα f, respectively, with some constantα= 0, then the solution of (1) does not change;

• iff andu_b are replaced byα f and α u_b, respectively, thenuchanges toα u;

• ifΩis transformed toF⁻¹(Ω) withF(x) =x/α, thenu◦F solves an analog of (1) inF⁻¹(Ω) with the data α²ε,αb◦F,c◦F,f◦F, and u_b◦F.

Note that the discrete problem (19) without the nonlinear termd_h preserves these properties. To preserve the properties also when using the nonlinear term, the functionγ_M has to satisfy

γ_M(ε,b, c, f, ub, Ω, u_h) =γ_M(α ε, αb, α c, α f, ub, Ω, u_h)

=α γ_M(ε,b, c, α f, α ub, Ω, α u_h)

=α⁻¹γ_F−1(M)(α²ε, αb◦F, c◦F, f◦F, u_b◦F, F⁻¹(Ω), u_h◦F)

for any admissible data,α= 0, andu_h∈W_h. We shall consider two choices of the scaling functionγ_M: a global one independent ofu_h and a local one depending onu_h. In the former case, one may set

γ_M =γ₀diam(Ω)^d/2

f_0,Ωdiam(Ω)

ε+b_0,∞,Ωdiam(Ω) +c_0,∞,Ωdiam(Ω)² + ub_0,∂Ω diam(Ω)^1/2

₋₁

(21)

with a positive constantγ₀. The local scaling can be defined by setting γ_M =β h^d/2_M /|uh|_1,M with a positive constantβ if|uh|_1,M = 0. Thus, we arrive at the following two formulas for the function ˜τ_M:

˜

τ_M =β h_M|bM|, (22)

and

˜

τ_M(u_h) =

⎧⎪

⎨

⎪⎩

β h^1+d/2_M |bM|

|uh|_1,M if|uh|_1,M = 0, 0 if|u_h|_1,M = 0,

(23) whereβ is a positive real number independent ofu_handh. The parameterβ depends on the data of (1) in case of (22) (e.g., like γ_M in (21)), but it is independent of the data of (1) in case of (23). For these two choices of ˜τ_M, we shall investigate the properties of the discrete problem (19). Although the local scaling is likely to lead to better numerical results than the global one, we consider both variants since the choice (22) turns out to be more appealing for the analysis.

Remark 3.1.

• Ifd= 2 and bM =0, one hasP_M =b^⊥_M⊗b^⊥_M where b^⊥_M is a vector satisfyingb^⊥_M ·bM = 0 and |b^⊥_M|= 1.

Thus, in this case, the nonlinear stabilization term can be written in the form d_h(w;u, v) =

M∈Mh

(τ_M^sold(w)κ_M(b^⊥_M · ∇u), κM(b^⊥_M · ∇v))M.

• It is useful for the analysis of the discrete problem to note thatκ_M(bM·∇u) =bM·κM∇uandκ_M(P_M∇u) = P_Mκ_M∇u. Note also thatPM₂= 1.

• Finally, if ˜τ_M is defined by (23), then, using the stability of κ_M and bM (18) and (16), respectively, and P_M2= 1, one obtains

τ_M^sold(v)_0,M ≤C h^1+d/2_M b_0,∞,M ∀v∈H¹(Ω), M ∈Mh. (24) In the analysis, the error will be measured using the following mesh-dependent norm

v_LPS:=

ε|v|²_1,Ω+σ^1/2v²_0,Ω+s_h(v, v) _1/2

,

and a term involving the crosswind derivative of the error. Note that integrating by parts gives

a(v, v) +s_h(v, v) =v²_LPS ∀v∈H₀¹(Ω). (25)

3.1. Well-posedness of the nonlinear discrete problem

This section studies the existence and uniqueness of solutions for the nonlinear discrete problem (19). The results of this section are valid also forσ₀= 0.

Let us define the nonlinear operatorT_h:V_h→V_h by

(T_hz_h, v_h) =a(z_h+u_bh, v_h) +s_h(z_h+u_bh, v_h) +d_h(z_h+u_bh;z_h+u_bh, v_h)−(f, v_h) (26) for anyz_h, v_h∈V_h. Then u_h∈W_h is a solution of (19) if and only ifu_h|_∂Ω =u_bh|_∂Ω and

T_h(u_h−u_bh) = 0,

or, equivalently,u_h=u_h+u_bh∈W_h is a solution of (19) ifu_h∈V_h andT_h(u_h) = 0.Thus, our aim is to prove that the operatorT_h has a zero inV_h. To this end, the properties of the formd_h shall be investigated first. As these properties are different with respect to the definition of ˜τ_M, we start supposing that ˜τ_M is given by (22).

Lemma 3.2. Let τ˜_M be defined by (22). Consider anyu, v, z∈W^1,3(Ω)and set w:=u−v. Then d_h(u;u, w)−d_h(v;v, w)≥1

7

M∈Mh

˜

τ_Mκ_M(P_M∇w)³_0,3,M =1

7d_h(w;w, w), (27)

|dh(u;u, z)−d_h(v;v, z)| ≤

M∈Mh

˜

τ_M(κM(P_M∇u)_0,3,M+κM(P_M∇v)_0,3,M)

× κM(P_M∇w)_0,3,MκM(P_M∇z)_0,3,M. (28) Proof. Let us denote

d_h(u;u, z)−d_h(v;v, z) =

M∈Mh

N_M(u, v, z), (29)

where

N_M(u, v, z) :=

τ_M^sold(u)κ_M(P_M∇u)−τ_M^sold(v)κ_M(P_M∇v), κM(P_M∇z) _M. Fort∈[0,1], letu^t:=tu+ (1−t)v and set

g(t) := ˜τ_M|κM(P_M∇u^t)|κ_M(P_M∇u^t), t∈[0,1].

Then

N_M(u, v, z) =

g(1)−g(0), κ_M(P_M∇z) _M = ₁

0

g(t) dt, κ_M(P_M∇z)

M

. Since

g(t) = ˜τ_M κ_M(P_M∇u^t)

|κM(P_M∇u^t)|κ_M(P_M∇u^t)·κ_M(P_M∇w) + ˜τ_M|κ_M(P_M∇u^t)|κ_M(P_M∇w), (30)

one has

|g(t)| ≤2 ˜τ_M|κM(P_M∇u^t)| |κM(P_M∇w)|

≤2 ˜τ_M(t|κM(P_M∇u)|+ (1−t)|κM(P_M∇v)|)|κM(P_M∇w)|,

which implies (28). On the other hand, since multiplication of the first term on the right-hand side of (30) by κ_M(P_M∇w) gives a non-negative expression, one obtains

N_M(u, v, w)≥

˜ τ_M

₁

0

|κM(P_M∇u^t)|dt κ_M(P_M∇w), κM(P_M∇w)

M

. (31)

Next, clearly

₁

0

|κM(P_M∇u^t)|dt≥ max

i=1,...,d

₁

0

|t κM(P_M∇u)i+ (1−t)κ_M(P_M∇v)i|dt.

Denoting

I(a, b) = ₁

0

|ta+ (1−t)b|dt, a, b∈R, a direct computation gives

I(a, b) = |a|+|b|

2 ifa b≥0, I(a, b) =1 2

a²+b²

|a|+|b| ifa b <0.

Thus, for anya, b∈R, it follows

I(a, b)≥ |a|+|b|

4 ≥ |a−b|

4 . Consequently,

₁

0

|κM(P_M∇u^t)|dt≥1 4 max

i=1,...,d|κM(P_M∇w)i| ≥ 1 4√

d|κM(P_M∇w)| ≥ 1

7|κM(P_M∇w)|.

Combining this estimate with (31) and using (29) gives (27).

Next, the properties ofd_hare explored for the case that ˜τ_M is defined by (23).

Lemma 3.3. Let τ˜_M be defined by (23). Consider anyu, v, z∈W^1,4(Ω). Then

|dh(u;v, z)| ≤C

M∈Mh

h^1+d/2_M b_0,∞,MκM(P_M∇v)_0,4,MκM(P_M∇z)_0,4,M, (32)

|dh(u;u, z)−d_h(v;v, z)| ≤C

M∈Mh

h^1+d/2_M b_0,∞,Mζ_M(u, v)×

×(κ_M(P_M∇u)_0,4,M+κ_M(P_M∇v)_0,4,M)κ_M(P_M∇z)_0,4,M, (33) where

ζ_M(u, v) =

⎧⎨

⎩

|u−v|_1,M

|u|_1,M+|v|_1,M if|u|_1,M = 0 or|v|_1,M = 0,

0 if|u|_1,M =|v|_1,M = 0.

Proof. Denoting

d_M(u;v, z) =

τ_M^sold(u)κ_M(P_M∇v), κ_M(P_M∇z) _M, it is easy to realize that

d_h(u;v, z) =

M∈Mh

d_M(u;v, z).

Applying H¨older’s inequality yields

|d_M(u;v, z)| ≤ τ_M^sold(u)_0,Mκ_M(P_M∇v)_0,4,Mκ_M(P_M∇z)_0,4,M, which, using (24), gives

|dM(u;v, z)| ≤C h^1+d/2_M b_0,∞,MκM(P_M∇v)_0,4,MκM(P_M∇z)_0,4,M, (34) thus proving (32). Now it will be shown that

|dM(u;u, z)−d_M(v;v, z)| ≤C h^1+d/2_M b_0,∞,Mζ_M(u, v)

×(κ_M(P_M∇u)_0,4,M+κ_M(P_M∇v)_0,4,M)κ_M(P_M∇z)_0,4,M. (35) If|u|_1,M = 0 or|v|_1,M = 0, then (35) is a particular case of (34). Thus, it suffices to consider the case|u|_1,M = 0,

|v|_1,M = 0. Denotingξ(x) =|x|x, one obtains

d_M(u;u, z)−d_M(v;v, z) =β h^1+d/2_M |bM|

|u|_1,M

ξ(κ_M(P_M∇u))−ξ(κ_M(P_M∇v)), κM(P_M∇z) _M

+β h^1+d/2_M |bM|

1

|u|_1,M − 1

|v|_1,M

ξ(κ_M(P_M∇v)), κM(P_M∇z) _M. (36) The integral terms on M possess the same structure as the term N_M(u, v, z) in the proof of Lemma 3.2(the second term corresponds to N_M(0, v, z)). They are estimated using the same technique, only with a different H¨older inequality. Then, (16) is applied to κM(P_M∇(u−v))_0,M resp. κM(P_M∇v)_0,M. Furthermore, the first inequality from (18) is employed. To finish the estimate of the second term in (36), the triangle inequality is used. One obtains

|dM(u;u, z)−d_M(v;v, z)| ≤C h^1+d/2_M b_0,∞,M |u−v|_1,M

|u|_1,M

×(κM(P_M∇u)_0,4,M+κM(P_M∇v)_0,4,M)κM(P_M∇z)_0,4,M.

The same type of inequality follows by interchanginguandv. Then, using the sharper of these two estimates and min{|u|⁻¹_1,M,|v|⁻¹_1,M} ≤2/(|u|1,M+|v|1,M) gives (35).

The properties of the operatorT_h, namely its monotonicity and local Lipschitz continuity, follow now by the results of the two previous lemmas and the representation of the LPS norm (25).

Lemma 3.4. If τ˜_M is defined by (22), then the operator T_h defined in (26) is locally Lipschitz-continuous and strongly monotone, i.e., it satisfies

(T_hw_h−T_hz_h, w_h−z_h)≥ w_h−z_h²_LPS+1 7

M∈Mh

˜

τ_Mκ_M(P_M∇(w_h−z_h))³_0,3,M (37) for allw_h, z_h∈V_h. Ifτ˜_M is defined by (23), then the operatorT_h is Lipschitz-continuous and it satisfies

(T_hz_h, z_h)≥ ε

2|zh|²_1,Ω−C₀(u_bh²_1,Ω+f²_0,Ω) (38) for allz_h∈V_h, whereC₀>0 depends onε,b, andc, but not on z_h,h, andσ₀.

Proof. Let us define the operatorsA_h, N_h:V_h→V_h by

(A_hz_h, v_h) =a(z_h, v_h) +s_h(z_h, v_h) ∀z_h, v_h∈V_h, (N_hz_h, v_h) =d_h(z_h+u_bh;z_h+u_bh, v_h) ∀ z_h, v_h∈V_h. Then, for anyw_h, z_h∈V_h, there holds

T_hw_h−T_hz_h=A_h(w_h−z_h) +N_hw_h−N_hz_h.

The operatorA_h is linear on a finite-dimensional space and hence it is Lipschitz continuous. Thus, the (local) Lipschitz-continuity ofT_h follows from (28), (33), and the equivalence of norms on finite-dimensional spaces.

The strong monotonicity (37) follows from (25) and (27). Finally, let ˜τ_M be defined by (23). In view of (25), it holds

(T_hz_h, z_h) =zh²_LPS+d_h(z_h+u_bh;z_h, z_h)

+a(u_bh, z_h) +s_h(u_bh, z_h) +d_h(z_h+u_bh;u_bh, z_h)−(f, z_h). (39) Applying (32), (10), (16), (18), (4), and (5), one obtains

|dh(z_h+u_bh;u_bh, z_h)| ≤C hb_0,∞,Ω|u_bh|_1,Ω|zh|_1,Ω.

The same estimate also holds for s_h(u_bh, z_h). Using the fact that d_h(z_h+u_bh;z_h, z_h) ≥ 0 and applying the Cauchy-Schwarz inequality to the third and last term on the right-hand side of (39), one derives

(T_hz_h, z_h)≥ε|z_h|²_1,Ω−(ε+Cb_0,∞,Ω+c_0,∞,Ω)u_bh1,Ωz_h1,Ω− f_0,Ωz_h0,Ω.

Now, employing the Poincar´e and Young inequalities, one obtains (38).

To prove that the discrete problem (19) has at least one solution, we shall use the following simple consequence of Brouwer’s fixed-point theorem, whose proof can be found in [32], p. 164, Lemma 1.4.

Lemma 3.5. LetX be a finite-dimensional Hilbert space with inner product(·,·)and norm·. LetP :X→X be a continuous mapping and K >0 a real number such that(P x, x)>0 for any x∈X with x=K. Then there existsx∈X such that x ≤K andP x= 0.

Collecting the previous results, the main result of this section can be stated now, namely, the well-posedness of the problem (19).

Theorem 3.6. If ˜τ_M is defined by (22) or (23), then the problem (19) has a solution. Ifτ˜_M is defined by (22), the solution of (19) is unique.

Proof. If ˜τ_M is defined by (22), then it follows from the strong monotonicity (37) that, for anyz_h∈V_h, (T_hz_h, z_h)≥ zh²_LPS+ (T_h0, z_h)≥ε|zh|²_1,Ω− Th0_0,Ωzh_0,Ω.

Thus, using Young’s inequality and the equivalence of norms in the spaceV_hone gets (T_hz_h, z_h)≥C₁zh²_0,Ω−C₂,

where C₁, C₂ are positive constants that depend onh and the data of (1), but not on z_h and σ₀. According to (38), the same inequality holds if ˜τ_M is defined by (23). Thus, in view of Lemma3.5with anyK >

C₂/C₁, the operatorT_h has a zero and hence the problem (19) has a solution. The uniqueness in the case that ˜τ_M is

defined by (22) follows from the strong monotonicity (37).

3.2. Error estimates

For the analysis of the methods introduced in Section3, we will need an appropriate interpolation operator.

An important tool for the construction of such an operator is provided by the following result, whose proof can be found in [25], Lemma 1.

Lemma 3.7. Let us suppose the inf-sup condition (9) to be satisfied. Then, there exists an operator _h : L²(Ω)→V_h such that, for any v, w∈L²(Ω), the estimates

|(v−_hv, w)| ≤C

M∈Mh

v_0,Mκ_Mw_0,M, (40)

|hv|²_1,M+h⁻²_M hv²_0,M ≤C

M∈Mh, M∩M=∅

h⁻²_Mv²_0,M ∀M ∈Mh (41)

are valid. Consequently, for any α∈R, it holds

M∈Mh

h^α_M(|hv|²_1,M+h⁻²_M hv²_0,M)≤C

M∈Mh

h^α−2_M v²_0,M, (42) where the constant C is independent ofv andhbut can depend on α.

With the operatorsi_hand _h, an operatorr_h∈L(C(Ω), W_h)∩L(C(Ω)∩H₀¹(Ω), V_h) is defined by

r_hv:=i_hv+_h(v−i_hv). (43)

To formulate the interpolation properties ofr_h, it is convenient to introduce the mesh dependent norm v_1,h=

M∈Mh

{|v|²_1,M +h⁻²_M v²_0,M} _1/2

.

Then, using (41), the geometrical hypotheses (4) and (5), and the approximation property ofi_h(11), one obtains v−r_hv_1,h≤Cv−i_hv_1,h≤C h˜ ^k|v|_k+1,Ω ∀v∈H^k+1(Ω), k= 1, . . . , l, (44) and consequently

|v−r_hv|_1,Ω+h⁻¹v−r_hv_0,Ω≤C h^k|v|_k+1,Ω ∀v∈H^k+1(Ω), k= 1, . . . , l. (45) The derivation of the error estimates will be based on the following two lemmas. The first one states an interpolation error estimate and the second one states a bound on the nonlinear formd_h.

Lemma 3.8. Let u∈H^k+1(Ω)for somek∈ {1, . . . , l}, and letη:=u−r_hu. Then, for anyv_h∈V_h\ {0}, the following estimate holds

η_LPS+a(η, v_h) +s_h(η, v_h)−s_h(u, v_h) v_hLPS

≤C

ε+hb_0,∞,Ω+h²σ_0,∞,Ω+h²|b|²_1,∞,Ωσ⁻¹₀ ^1/2h^k|u|_k+1,Ω. (46) Proof. Since, in view of (5), (16), (18), and the definition ofτ_M (20)

v_LPS≤C

ε+hb_0,∞,Ω+h²σ_0,∞,Ω ^1/2v_1,h ∀v∈H¹(Ω),